1D Dirac equation · topological mass kink · zero-energy bound state · mid-gap protection
Jackiw and Rebbi (1976) showed that a 1D Dirac fermion in a spatially varying mass field m(x) = m₀ tanh(x/ξ) — a topological kink — necessarily has a zero-energy bound state localized at the kink center. The Hamiltonian is:
H = -iℏ σ_z ∂_x + m(x) σ_x
The zero-mode wavefunction: ψ₀(x) ∝ exp(−∫₀ˣ m(x')dx') = sech(x/ξ)^(m₀ξ)
This is the prototype of topological protection: the mid-gap state cannot be removed by any continuous deformation that preserves the kink topology. The topological charge Q = [sgn(m(+∞)) − sgn(m(−∞))]/2 = 1 counts the number of zero modes (index theorem). Physical realizations: polyacetylene solitons (SSH model), Majorana zero modes in topological superconductors, graphene domain walls.